# nLab infinitary tensor product

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

## Applications

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

#### Limits and colimits

limits and colimits

# Contents

## Idea

In a category with all products, one can take finite products of objects, and hence one has a monoidal category. One can however also take infinite products. In some situations, such as in categorical probability, one is interesting in taking a infinitary version of a tensor product.

In measure-theoretic probability theory, such a construction is provided by Kolmogorov's extension theorem, and the same idea can be used in general:

• In a category with products, an infinite product can be expressed as a cofiltered limit of finite products.
• Similarly, one can define an infinitary tensor product as a cofiltered limit, if it exists, of finite tensor products.

## Definition

Let $(C,\otimes,I)$ be a symmetric monoidal category equipped with a distinguished map $del_X:X\to I$ for each object $I$. (This happens for example if $C$ is cartesian semicartesian, or if it has a Markov or copy-discard structure.)

Let $J$ be a possibly infinite set, and let $(X_i)_{i\in J}$ be a $J$-indexed collection of objects of $C$.

For every finite subset $F\subseteq J$, denote by

$\bigotimes_{i\in F} X_i$

the tensor product of the family $(X_i)_{i\in F}$. (Note that $F$ is not ordered, but the ordering of the terms in the tensor product does not matter up to coherent isomorphism, since $C$ is symmetric monoidal.)

Consider now finite subsets $F\subseteq G$ of $J$. We have a canonical “projection” morphism defined as follows. First of all, for $i\in G$, let

$Y_i = \begin{cases} X_i & i\in F ; \\ I & i\notin F. \end{cases}$

Similarly, let $e_i:X_i\to Y_i$ be

$e_i = \begin{cases} id_{X_i} & i\in F ; \\ del_{X_i} & i\notin F. \end{cases}$

The map $\pi_{G,F}$ is given by the tensor product of the $e_i$ as follows, where the isomorphism on the right is given by the unitors of the monoidal category.

Let now $Fin(J)$ be the poset of finite subsets of $G$. The union of finite subsets is finite, so $Fin(J)$ is a directed set, hence a filtered category. The functor $Fin(J)^op\to C$ mapping the inclusion $F\subseteq G$ to $\pi_{G,F}$ defined above is hence a cofiltered diagram.

We say that an object is an infintary tensor product of the family $(X_i)_{i\in J}$ in $C$, and denote it by $X_J$, if

• It is a cofiltered limit of the functor $Fin(J)^op\to C$ described above, and moreover
• For every object $A$, the functor $A\otimes-:C\to C$ preserves this limit.

## Examples

• In a cartesian monoidal category, every infinite product, if it exists, is an infinitary tensor product.

• In a Markov category, Kolmogorov products are particular infinitary products compatible with the copy-discard structure. In particular, the category BorelStoch has all countable infinitary tensor products.

• Reversing all arrows, the infinite tensor product of rings is defined as the filtered colimit of all the finitary tensor products. (The canonical arrows $I\to X$ are the units of the rings.)